Q26P - Fundamentals Of Physics

Question

A muon (an elementary particle) enters a region with a speed of $\mathbf{5}\mathbf{.}\mathbf{00}\mathbf{\times }{\mathbf{10}}^{\mathbf{6}}\mathbf{ }\mathbf{m}\mathbf{/}\mathbf{s}$ and then is slowed down at the rate of $\mathbf{1}\mathbf{.}\mathbf{25}\mathbf{\times }{\mathbf{10}}^{\mathbf{14}}\mathbf{ }\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}$. (a) How far does the muon take to stop? (b) Graph x vs t and v vs t for the muon.

Step-by-Step Solution

Answer

(a) The distance muon takes to stop is 0.100 m .

(b) The graphs of x vs t and v vs tare plotted for the muon.

Step 1: Given Data

Initial speed of muon is, $v_i=5.00\times {10}^6 \mathrm{m}/\mathrm{s}$

Acceleration of the muon is, $a=-1.25\times {10}^{14} \mathrm{m}/{\mathrm{s}}^2$

Step 2: Understanding the kinematic equations of motion.

Kinematic equations of motion describe the relationship between displacement, velocity or acceleration of a particle. The graphs of displacement vs time and velocity vs timecan be plotted to analyze the motion of the particle.

The expression for the equation of motion is given as below,

$v_f^2=v_i^2+2ax$(i)

Here, $v_f$ is the final velocity, $v_i$ is the initial velocity, $a$ is the acceleration, and $x$ is the distance.

Step3: (a) Determination of the distance required for muon to stop

Rearrange the equation (i) for x and substitute the values.

$$\begin{gathered} x=\frac{v_f^2-v_i^2}{2a} \\ =\frac{\left(0^2-{\left(5.00\times {10}^6\right)}^2\right)}{\left(2\times \left(-1.25\right)\times {10}^{14}\right)} \\ =0.100 \mathrm{m} \end{gathered}$$

Step 4: (b) Plotting the graph of x vs t and v vs t .

To plot graph of vs kinematic equationis used,

$$\begin{gathered} x=v_{i}t+\frac{1}{2}a{t}^2 \\ x=\left(5.00\times {10}^6\times t\right)+\left(\frac{1}{2}\times \left(-1.25\times {10}^{14}\times t^2\right)\right) \end{gathered}$$

To plot graph of vs t kinematic equation is used,

$$\begin{gathered} v=v_i+at \\ v=\left(5.00\times {10}^6\right)+\left(-1.25\times {10}^{14}\times t\right) \end{gathered}$$